19. A note on the product of element orders of finite groups with a Sylow tower (with C. Monetta)

submitted

Abstract: Let G be a finite group and denote by r(G) the product of its element orders. In this note, we find a bound for r(G) in terms of r(Cn), where G is a non-cyclic group of order n admitting a Sylow tower.

18. Applications of Automaton Groups in Cryptography (with D. Kahrobaei and E. Rodaro)

submitted, available at https://arxiv.org/pdf/2306.10522.pdf

Abstract: In 1991 the first public key protocol involving automaton groups has been proposed. In this paper, we give a survey about algorithmic problems around automaton groups which may have potential applications in cryptography. We then present a new public key protocol based on the conjugacy search problem in some families of automaton groups. At the end, we offer open problems that could be of interest to group theorists and computer scientists in this direction.

17. Poweful 3-Engel groups (with I. de las Heras and G. Traustason)

submitted, available at https://arxiv.org/abs/2301.070

Abstract: In this paper we study powerful 3-Engel groups. In particular, we find sharp upper bounds for the nilpotency class of powerful 3-Engel groups and the subclass of powerful metabelian 3-Engel groups.

16. On the structure of finite groups determined by the arithmetic and geometric means of element orders (with V. Grazian and C. Monetta)

submitted, available at https://arxiv.org/pdf/2212.13770.pdf

Abstract: In this paper we consider two functions related to the arithmetic and geometric means of element orders of a finite group, showing that certain lower bounds on such functions strongly affect the group structure. In particular, for every prime p, we prove a sufficient condition for a finite group to be p-nilpotent, that is, a group whose elements of p'-order form a normal subgroup.

15. A unified description of cryptosystems: from classical to quantum protocols (with G. De Riso)

Journal of Discrete Mathematical Sciences & Cryptography, vol. 26 no. 8 (2023) pp. 2285-2299

Abstract: The launch of quantum computing has created new challenges in the field of cryptography. Nowadays this field of research is extremely active in physics, mathematics and computer science. In this paper, we present an overview of the most common classical and quantum protocols. The former are usually defined through a 5-uple given by a plaintext space, a ciphertext space, an encryption and decryption space, an encryption function and a decryption function. Here, we show how to unify quantum and classical protocols by using the same formalism of the 5-uple mentioned above.

14. Upper bounds for the product of element orders of finite groups (with E. Di Domenico and C. Monetta)

Journal of Algebraic Combinatorics, vol. 57 no. 4 (2023) pp. 1033-1043

Abstract: Let G be a finite group of order n, and denote by r(G) the product of element orders of G. The aim of this work is to provide some upper bounds for r(G) depending only on n and on its least prime divisor, when G  belongs to some classes of non-cyclic groups.

13. The root extraction problem in braid group-based cryptography (with M. Cumplido and D. Kahrobaei)

submitted, available at https://arxiv.org/pdf/2203.15898.pdf

Abstract: The root extraction problem in braid groups is the following: given a braid b in Bn and a natural number k, find a in Bn such that a^k=b. In the last decades, many cryptosystems  such as authentication schemes and digital signatures  based on the root extraction problem have been proposed. In this paper, we first describe these cryptosystems built around braid groups. Then we prove that, in general, these authentication schemes and digital signature are not secure by presenting for each of them a possible attack.

12. Tree languages and branched groups (with L. Bartholdi)

Mathematische Zeitschrift, vol. 303 (4) no. 96 (2023)

Abstract: We study the portraits of isometries of rooted trees - the labelling of the tree, at each vertex, by the permutation of its descendants - in terms of languages. We characterize regularly branched self-similar groups in terms of ω-regular languages. We deduce the algorithmic decidability of some problems, such as the comparison of regularly branched contracting groups, and their orbit structure on the boundary of the rooted tree.

11. Group-based Cryptography in the Quantum Era (with D. Kahrobaei and R. Flores)

Notices of the American Mathematical Society, vol. 70 no. 5 (2023) pp. 2-13

Abstract: In this expository article we present an overview of the current state-of-the-art in Group-based cryptography, with an eye in possible candidates for Post-Quantum cryptography (PGC). We describe several families of groups that have been proposed as platforms, with special emphasis in polycyclic groups and graph groups, dealing in particular with their algorithmic properties and cryptographic applications. We also deal with fully homomorphic encryption, which enables computation with encrypted data. We end up by discussing several open problems in the field.

10. p-Basilica groups (with  E. Di Domenico, G.A. Fernández-Alcober, and A. Thillaisundaram)

Mediterranean Journal of Mathematics, vol. 19 no. 275 (2022) pp. 1-28

Abstract: We consider a generalisation of the Basilica group to all odd primes: the p-Basilica groups acting on the p-adic tree. We show that the p-Basilica groups have the p-congruence subgroup property but not the congruence subgroup property nor the weak congruence sub- group property. This provides the first examples of weakly branch groups with such properties. In addition, the p-Basilica groups give the first examples of weakly branch, but not branch, groups which are super strongly fractal. We compute the orders of the congruence quotients of these groups, which enable us to determine the Hausdorff dimensions of the p-Basilica groups. Lastly, we show that the p-Basilica groups do not possess maximal subgroups of infinite index and that they have infinitely many non-normal maximal subgroups.

9. Ramification structures for quotients of the Grigorchuk groups (with A. Thillaisundaram)

Journal of Algebra and its Applications, vol. 22 no. 2 (2023) pp. 1037-10

Abstract: Groups of surfaces isogenous to a higher product of curves can be characterised by a purely group-theoretic condition, which is the existence of a so-called ramification structure. In this paper, we prove that infinitely many quotients of the Grigorchuk groups admit ramification structures. This gives the first explicit infinite family of 3-generated finite 2-groups with ramification structures that are not Beauville.

8. Locally finite p-groups with a left 3-Engel element whose normal closure is not nilpotent (with A. Hadjievangelou and G. Traustason)

International Journal of Algebra and Computation, vol. 31 no. 1 (2020) pp. 135-160

Abstract: For any odd prime p, we give an example of a locally finite p-group G containing a left 3-Engel element x whose normal closure is not nilpotent.

7. A family of fractal non-contracting weakly branch groups

Ars Mathematica Contemporanea, vo.l 20 no. 1 (2021)

Abstract: We construct a new example of an infinite family of groups acting on a d-adic tree, with d>1 that is non-contracting and weakly regular branch over the derived subgroup. 

6. Hausdorff dimension of the second Grigorchuk group (with A. Thillaisundaram)

International Journal of Algebra and Computation, vol. 31 no. 06 (2021) pp. 1037-1047

Abstract: We show that the Hausdorff dimension of the closure of the second Grigorchuk group is 43/128. Furthermore we establish that the second Grigorchuk group is super strongly fractal and that its automorphism group equals its normaliser in the full automorphism group of the tree.

5. Algorithmic problems in Engel groups and cryptographic applications (with D. Kahrobaei)

International Journal of Group Theory, vol. 9 no. 4 (2020) pp. 231-250

Abstract: The theory of Engel groups plays an important role in group theory since these groups are closely related to the Burnside problems. In this survey we consider several classical and novel algorithmic problems for Engel groups and propose several open problems. We study these problems with a view towards applications to cryptography.

4. Engel elements in weakly branch groups (with G. A. Fernández Alcober and G. Tracey)

Journal of Algebra, vol. 554 (2020) pp. 54-77

Abstract: We study properties of Engel elements in weakly branch groups, lying in the group of automorphisms of a spherically homogeneous rooted tree. More precisely, we prove that the set of bounded left Engel elements is always trivial in weakly branch groups. In the case of branch groups, the existence of non-trivial left Engel elements implies that these are all p-elements and that the group is virtually a p-group (and so periodic) for some prime p. We also show that the set of right Engel elements of a weakly branch group is trivial under a relatively mild condition. Also, we apply these results to well-known families of weakly branch groups, like the multi-GGS groups.

3. A left 3-Engel element whose normal closure is not nilpotent (with G. Tracey and G. Traustason)

Journal of Pure and Applied Algebra, vol. 224 no. 3 (2020) pp. 1092-1101

Abstract: We give an example of a locally nilpotent group G containing a left 3-Engel element x where its normal closure is not nilpotent.

2. Engel elements in some fractal groups (with G. A. Fernández Alcober and A. Garreta)

Monatshefte für Mathematik, vol. 189 no. 4 (2019) pp. 651-660

Abstract: Let p be a prime and let G be a subgroup of a Sylow pro-p subgroup of the group of automorphisms of the p-adic tree. We prove that if G is fractal and |G′: st(1)′| is not finite, then the set L(G) of left Engel elements of G is trivial. This result applies to fractal nonabelian groups with torsion-free abelianization, for example the Basilica group, the Brunner-Sidki-Vieira group, and also to the GGS-group with constant defining vector. We further provide two examples showing that neither of the requirements |G′: st(1)′| = ∞ and being fractal can be dropped.

1. A note on Engel elements in the first Grigorchuk group (with A. Tortora)

International Journal of Group Theory, vol. 8 no. 3 (2019) pp. 9-14

Abstract: Let Γ be the first Grigorchuk group. According to a result of Bartholdi, the only left Engel elements of Γ are the involutions. This implies that the set of left Engel elements of Γ is not a subgroup. Of particular interest is to wonder whether this happens also for the sets of bounded left Engel elements, right Engel elements, and bounded right Engel elements of Γ. Motivated by this, we prove that these three subsets of Γ coincide with the identity subgroup.